Finite Group Databasebeta

← C3×#81602 #11878701

Subgroups — C3×#81602 #11878701

Order 1152.

905 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5725C211yesmaximal, characteristic
2C3×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
3C32×#487C211yesmaximal, characteristic, complemented
4C3×#5093C211yesmaximal, characteristic, complemented
5C3×#5085C211yesmaximal, characteristic, complemented
6C3×#4818C214yesmaximal, complemented
7C3×#4806C214yesmaximal, complemented
8C3×#4693C211yesmaximal, characteristic, complemented
9C3×#4685C211yesmaximal, characteristic, complemented
10C3×#3206—31nomaximal
11#81602C311yesmaximal, characteristic, direct product factor
12C2×C3×#239—21no—
13C2×C3×D3×Q8C2212yes—
14C32×C4⋊Q8C2211yescharacteristic
15C32×C4⋊Q8C2211yescharacteristic
16C3×#947—21no—
17C3×#944C2212yes—
18C2×C3×#215C2214yes—
19C2×C3×#204C2214yes—
20C3×#895—21no—
21C3×#895C2212yes—
22C3×#893—21no—
23C3×#893C2212yes—
24C3×D3×C4⋊C4—21no—
25C3×D3×C4⋊C4—21no—
26C3×#888—21no—
27C3×#888—21no—
28C3×#874C2211yescharacteristic
29C3×C42×D3C2211yescharacteristic
30C3×#869C2211yescharacteristic
31C32×#103C2214yes—
32C32×C4×C8C2211yescharacteristic
33C3×(C2×D3)⋊C8C2212yes—
34C3×Dic3⋊C8C2212yes—
35C3×#810C2214yes—
36C3×C4×C3⋊C8C2211yescharacteristic
37C3×#532—31no—
38C2×C3×C4⋊Q8—31no—
39#5725C2×C311yescharacteristic
40D3×C4⋊Q8C2×C311yescharacteristic, complemented
41C3×#492—31no—
42C3×#492—31no—
43C3×#487—21no—
44C3×#487—31no—
45C3×#487D311yescharacteristic, complemented
46C3×#487C2×C311yescharacteristic, complemented
47C3×#461—34no—
48C3×#448—34no—
49C3×#431—31no—
50#5093C2×C311yescharacteristic, complemented
51#5085C2×C311yescharacteristic, complemented
52#4818C2×C314yescomplemented
53#4806C2×C314yescomplemented
54#4693C2×C311yescharacteristic, complemented
55#4685C2×C311yescharacteristic, complemented
56C2×C32×Q8—21no—
57C2×C32×Q8C2312yes—
58C3×#239—44no—
59C3×D3×Q8—44no—
60C2×C3×D12—22no—
61C2×C3×D12—21no—
62C2×C3×D12—21no—
63C2×C3×D12C2312yes—
64C2×C3×C4×D3—22no—
65C2×C3×C4×D3—22no—
66C2×C3×C4×D3D411yescharacteristic
67C2×C3×C4×D3D411yescharacteristic
68C2×C3×C4×D3C2311yescharacteristic
69C2×C3×Dic6—22no—
70C2×C3×Dic6C2312yes—
71C2×C32×C8C2312yes—
72C32×C4⋊C4—21no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4C2312yes—
76C32×C42C2311yescharacteristic
77C3×#215—44no—
78C3×(C2×D3)⋊C4—22no—
79C3×(C2×D3)⋊C4—22no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—21no—
82C3×Dic3⋊C4—22no—
83C3×Dic3⋊C4—22no—
84C3×C4×Dic3D411yescharacteristic
85C3×C4×Dic3D411yescharacteristic
86C3×C4×Dic3C2311yescharacteristic
87C2×C3×C3⋊C8C2312yes—
88C3×#204—44no—
89#3206—31noSylow
90C2×C3×#41—61no—
91C22×C3×Q8—32no—
92C2×#239—21no—
93C2×D3×Q8C22×C312yes—
94C2×C3×SD16—34no—
95C2×C3×SD16—34no—
96C3×C4⋊Q8—21no—
97C3×C4⋊Q8—21no—
98C3×C4⋊Q8—32no—
99C3×C4⋊Q8—31no—
100C3×C4⋊Q8—61no—
101C3×C4⋊Q8—61no—
102C3×C4⋊Q8C2×D311yescharacteristic
103C3×C4⋊Q8C2×D311yescharacteristic
104C3×C4⋊Q8C22×C311yescharacteristic
105C3×C4⋊Q8C22×C311yescharacteristic
106C3×#127—31no—
107C3×#124—61no—
108C3×#121—32no—
109C3×#121—61no—
110C3×C4×D4—32no—
111C3×C4×D4—61no—
112C2×C3×C4⋊C4—61no—
113C2×C3×C4⋊C4—61no—
114C2×C3×C42—31no—
115#947—21no—
116#944C22×C312yes—
117C2×#215C22×C314yes—
118C2×#204C22×C314yes—
119#895—21no—
120#895C22×C312yes—
121#893—21no—
122#893C22×C312yes—
123D3×C4⋊C4—21no—
124D3×C4⋊C4—21no—
125#888—21no—
126#888—21no—
127#874C22×C311yescharacteristic
128C42×D3C22×C311yescharacteristic
129#869C22×C311yescharacteristic
130C3×C4⋊C8—32no—
131C3×#103—24no—
132C3×#103—34no—
133C3×#103C2×D314yes—
134C3×#103C22×C314yes—
135C3×C22⋊C8—32no—
136C3×C4×C8—21no—
137C3×C4×C8—31no—
138C3×C4×C8C2×D311yescharacteristic
139C3×C4×C8C22×C311yescharacteristic
140(C2×D3)⋊C8C22×C312yes—
141Dic3⋊C8C22×C312yes—
142#810C22×C314yes—
143C4×C3⋊C8C22×C311yescharacteristic
144C22×C3×D3—22no—
145C22×C3×D3C2×D411yescharacteristic
146C32×Q8—24no—
147C32×Q8—42no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2411yescharacteristic
153C2×C3×Dic3—22no—
154C2×C3×Dic3C2×D411yescharacteristic
155C2×C3×Dic3C2×D411yescharacteristic
156C2×C3×Dic3C2×D411yescharacteristic
157C3×D12—24no—
158C3×D12—44no—
159C3×D12—42no—
160C3×D12—42no—
161C3×C4×D3—22no—
162C3×C4×D3—22no—
163C3×C4×D3—44no—
164C3×C4×D3—44no—
165C3×C4×D3SD1614yescomplemented
166C3×Dic6—24no—
167C3×Dic6—44no—
168C32×C8—22no—
169C3×C3⋊C8—22no—
170#532—31no—
171C2×C4⋊Q8—31no—
172#492—31no—
173#492—31no—
174#487—31no—
175#487C3×D311yescharacteristic, complemented
176#461—34no—
177#448—34no—
178#431—31no—
179C3×#41—124no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—22no—
183C2×C3×Q8—32no—
184C2×C3×Q8—41no—
185C2×C3×Q8—62no—
186C2×C3×Q8—124no—
187C2×C3×Q8C22×D312yes—
188C2×C3×Q8C23×C312yes—
189C2×C3×D4—32no—
190C2×C3×D4—62no—
191C2×C3×D4—61no—
192C2×C3×D4—61no—
193C22×C3×C4—31no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—62no—
197C22×C3×C4—62no—
198#239—44no—
199D3×Q8—44no—
200C2×D12—22no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12C23×C312yes—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3C3×D411yescharacteristic
207C2×C4×D3C3×D411yescharacteristic
208C2×C4×D3C23×C311yescharacteristic
209C2×Dic6—22no—
210C2×Dic6C23×C312yes—
211C3×SD16—124no—
212C3×SD16—124no—
213C2×C3×C8—22no—
214C2×C3×C8—32no—
215C2×C3×C8C22×D312yes—
216C2×C3×C8C23×C312yes—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—41no—
226C3×C4⋊C4—41no—
227C3×C4⋊C4—62no—
228C3×C4⋊C4—62no—
229C3×C4⋊C4—61no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4C22×D312yes—
232C3×C4⋊C4C23×C312yes—
233C3×C22⋊C4—62no—
234C3×C22⋊C4—62no—
235C3×C42—21no—
236C3×C42—31no—
237C3×C42—31no—
238C3×C42—31no—
239C3×C42C22×D311yescharacteristic
240C3×C42C23×C311yescharacteristic
241#215—44no—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248C4×Dic3C3×D411yescharacteristic
249C4×Dic3C3×D411yescharacteristic
250C4×Dic3C23×C311yescharacteristic
251C2×C3⋊C8C23×C312yes—
252#204—44no—
253C22×C32C22×D411yescharacteristic
254C2×C3×D3—21no—
255C2×C3×D3—21no—
256C2×C3×D3—42no—
257C2×C3×D3—42no—
258C2×C3×D3—42no—
259C2×C3×D3C2×SD1612yes—
260C32×C4—42no—
261C32×C4—42no—
262C32×C4C2×SD1612yes—
263C32×C4#13712yes—
264C32×C4H(5,𝔽2)12yes—
265C3×Dic3—21no—
266C3×Dic3—21no—
267C3×Dic3—42no—
268C3×Dic3C2×SD1612yes—
269C2×#41—61no—
270C22×Q8—32no—
271C2×SD16—34no—
272C2×SD16—34no—
273C4⋊Q8—32no—
274C4⋊Q8—31no—
275C4⋊Q8—61no—
276C4⋊Q8—61no—
277C4⋊Q8C2×C3×D311yescharacteristic
278C4⋊Q8C2×C3×D311yescharacteristic
279#127—31no—
280#124—61no—
281#121—32no—
282#121—61no—
283C4×D4—32no—
284C4×D4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C42—31no—
288C4⋊C8—32no—
289#103—34no—
290#103C2×C3×D314yes—
291C22⋊C8—32no—
292C4×C8—31no—
293C4×C8C2×C3×D311yescharacteristic
294C23×C3—31no—
295C23×C3—62no—
296C22×D3—22no—
297C22×D3C2×C3×D411yescharacteristic
298C3×Q8—24no—
299C3×Q8—24no—
300C3×Q8—44no—
301C3×Q8—42no—
302C3×Q8—42no—
303C3×Q8—64no—
304C3×Q8—82no—
305C3×Q8—124no—
306C3×D4—64no—
307C3×D4—124no—
308C3×D4—122no—
309C3×D4—122no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—31no—
318C2×C3×C4—34no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—124no—
327C2×C3×C4—124no—
328C2×C3×C4D3×D411yescharacteristic
329C2×C3×C4D3×D411yescharacteristic
330C2×C3×C4C2×C3×D411yescharacteristic
331C2×C3×C4C2×C3×D411yescharacteristic
332C2×C3×C4C23×D311yescharacteristic
333C2×C3×C4C24×C311yescharacteristic
334C2×Dic3—22no—
335C2×Dic3C2×C3×D411yescharacteristic
336C2×Dic3C2×C3×D411yescharacteristic
337C2×Dic3C2×C3×D411yescharacteristic
338D12—24no—
339D12—44no—
340D12—42no—
341D12—42no—
342C4×D3—22no—
343C4×D3—22no—
344C4×D3—44no—
345C4×D3—44no—
346C4×D3C3×SD1614yescomplemented
347Dic6—24no—
348Dic6—44no—
349C3×C8—22no—
350C3×C8—22no—
351C3×C8—42no—
352C3×C8—62no—
353C3⋊C8—22no—
354C2×C32#53511yescharacteristic
355C2×C32C22×SD1611yescharacteristic
356C2×C32C2×#13711yescharacteristic
357C3×D3—22no—
358C3×D3—82no—
359#41—124no—
360C2×Q8—21no—
361C2×Q8—32no—
362C2×Q8—62no—
363C2×Q8—124no—
364C2×Q8C22×C3×D312yes—
365C2×D4—32no—
366C2×D4—62no—
367C2×D4—61no—
368C2×D4—61no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—62no—
373C22×C4—62no—
374SD16—124no—
375SD16—124no—
376C2×C8—32no—
377C2×C8C22×C3×D312yes—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—61no—
384C4⋊C4—61no—
385C4⋊C4C22×C3×D312yes—
386C22⋊C4—62no—
387C22⋊C4—62no—
388C42—31no—
389C42—31no—
390C42—31no—
391C42C22×C3×D311yescharacteristic
392C22×C3—21no—
393C22×C3—32no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—122no—
397C22×C3—122no—
398C22×C3—122no—
399C22×C3C2×D3×D411yescharacteristic
400C22×C3C22×C3×D411yescharacteristic
401C2×D3—21no—
402C2×D3—21no—
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3C2×C3×SD1612yes—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—32no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—61no—
416C3×C4—61no—
417C3×C4—82no—
418C3×C4—82no—
419C3×C4—122no—
420C3×C4D3×SD1612yes—
421C3×C4#91812yes—
422C3×C4C2×C3×SD1612yes—
423C3×C4C3×#13712yes—
424C3×C4#100912yes—
425C3×C4C3×H(5,𝔽2)12yes—
426Dic3—21no—
427Dic3—21no—
428Dic3—42no—
429Dic3C2×C3×SD1612yes—
430C32#320611yescharacteristic, complemented, Sylow
431C23—31no—
432C23—62no—
433Q8—24no—
434Q8—42no—
435Q8—64no—
436Q8—124no—
437D4—64no—
438D4—124no—
439D4—122no—
440D4—122no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—31no—
444C2×C4—34no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—124no—
451C2×C4—124no—
452C2×C4C3×D3×D411yescharacteristic
453C2×C4C3×D3×D411yescharacteristic
454C2×C4C23×C3×D311yescharacteristic
455C8—22no—
456C8—62no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—62no—
461C2×C3—242no—
462C2×C3#567411yescharacteristic
463C2×C3C2×D3×SD1611yescharacteristic
464C2×C3C2×#91811yescharacteristic
465C2×C3C3×#53511yescharacteristic
466C2×C3C22×C3×SD1611yescharacteristic
467C2×C3C2×C3×#13711yescharacteristic
468D3—22no—
469D3—82no—
470C22—32no—
471C22—61no—
472C22—61no—
473C22—122no—
474C22—122no—
475C22—122no—
476C22C2×C3×D3×D411yescharacteristic
477C4—32no—
478C4—42no—
479C4—42no—
480C4—61no—
481C4—61no—
482C4—122no—
483C4C3×D3×SD1612yes—
484C4C3×#91812yes—
485C4C3×#100912yes—
486C3—21no—
487C3#8160211yescharacteristic, direct product factor
488C3C3×#320611yescharacteristic, complemented
489C2—62no—
490C2—242no—
491C2C3×#567411yescharacteristic
492C2C2×C3×D3×SD1611yescharacteristic
493C2C2×C3×#91811yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#487C2
socleC22×C32C22×D4
prime radicalC3×#5725C2
hypercenterC3×#487D3
hypocenterC3C3×#3206
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#487

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×D3×C4⋊Q8C2semidirect
3. C32×#487C2semidirect
4. C3×#5093C2semidirect
5. C3×#5085C2semidirect
6. C3×#4818C2semidirect
7. C3×#4806C2semidirect
8. C3×#4693C2semidirect
9. C3×#4685C2semidirect
11. #81602C3direct (N × H)
40. D3×C4⋊Q8C2×C3semidirect
45. C3×#487D3semidirect
46. C3×#487C2×C3semidirect
50. #5093C2×C3semidirect
51. #5085C2×C3semidirect
52. #4818C2×C3semidirect
53. #4806C2×C3semidirect
54. #4693C2×C3semidirect
55. #4685C2×C3semidirect
165. C3×C4×D3SD16semidirect
175. #487C3×D3semidirect
346. C4×D3C3×SD16semidirect
430. C32#3206semidirect
487. C3#81602direct (N × H)
488. C3C3×#3206semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#57252
C3×D3×C4⋊Q82
C32×#4872
C3×#50932
C3×#50852
C3×#48182
C3×#48062
C3×#46932
C3×#46852
C3×#32063
#816023